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In his book Introduction to Metric and Topological Spaces, author Wilson A Sutherland in explaining the equivalence of metrics invoked the definition:

Two metrics $d_1, d_2$ on a set $X$ will be called Lipschitz equivalent if there are positive constants $h, k$ such that for any $x, y \in X, $ $$ hd_2(x, y) \leq d_1(x, y) \leq kd_2(x, y) .$$ The reader should be warned that although the name we have chosen for the concept seems appropriate, it is not universally used.

It's a familiar definition that is used to show equivalence of metrics and in fact it is equivalent to topological equivalence. However, prior Sutherland, I was not familiar with the name attributed to Lipschitz. I googled and tried to narrow down any specific paper that he wrote concerning the equivalence thus lending his name, however of no avail.

So, my question is, did Lipschitz formulate the equivalence himself or were it some authors who attributed the equivalence to him in their treatises?

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    $\begingroup$ I think Rudolf Lipschitz used functions on $\mathbb R^n$ with the property $|f(x)-f(y)| \le M|x-y|$ for some constant $M$. This property of functions has come to bear his name. Lipschitz himself likely never mentioned metric space, nor the word "equivalent". As Sutherland noted, this is not standard terminology for metrics even now. $\endgroup$ Aug 24, 2022 at 10:26
  • $\begingroup$ To be fair @GeraldEdgar, this is good assessment. From the undergrad days, we are introduced to Lipschitz functions. However, this was alien to me. If Lipschitz wasn't instrumental himself, then I am fascinated who first started to use this term. $\endgroup$ Aug 24, 2022 at 10:36
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    $\begingroup$ Lipschitz condition (see comment by G. Edgar) first appeared in a print as a sufficient condition for convergence of a Fourier series of function $f(x)$: R. Lipschitz, "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, et praecipue earum, quae per variabilis spatium finitum valorum maximourm et minimorum numerum habent infinitum, disquisitio." Journal für die reine und angewandte Mathematik, Vol. 63, 1864, pp. 296-308. (scan) $\endgroup$
    – njuffa
    Aug 24, 2022 at 12:19
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    $\begingroup$ The earliest publication I could find that refers to the Lipschitz condition by that name ("Lipschitz'sche Bedingung") dates to 1877. Metric spaces and Lipschitz equivalent are not in the picture at at time. The earliest relevant publications are from the 1960s, e.g. Robert B. Fraser, "A new metric for a metric space", Proceedings of the American Mathematical Society, Vol. 21, No. 3, June 1969, pp. 755-761: " Two metric spaces are called Lipschitz equivalent if there is a function from one to the other such that both the function and its inverse satisfy a local Lipschitz condition [..] " $\endgroup$
    – njuffa
    Aug 24, 2022 at 12:27
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    $\begingroup$ @User1865345 Not necessarily. This is the one unambiguous source from the 1960s for which I could access the full text. I was unable to find any instances of "Lipschitz equivalent" in publications from the 1950s or earlier. $\endgroup$
    – njuffa
    Aug 24, 2022 at 19:14

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